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discuss proper compat measure

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18
latex/tex/appendix.tex
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@ -89,6 +89,24 @@ uniformly distributed samples into samples distributed like \(\rho\).
Given a variable transformation, one can reconstruct a corresponding
probability density, by chaining the Jacobian with the inverse of that
transformation.
\subsection{Compatibility of Histograms}
\label{sec:comphist}
The compatibility of histograms is tested as described
in~\cite{porter2008:te}. The test value
is \[T=\sum_{i=1}^k\frac{(u_i-v_i)^2}{u_i+v_i}\] where \(u_i, v_i\)
are the number of samples in the \(i\)-th bins of the histograms
\(u,v\) and \(k\) is the number of bins. This value is \(\chi^2\)
distributed with \(k\) degrees, when the number of samples in the
histogram is reasonably high. The mean of this distribution is \(k\)
and its standard deviation is \(\sqrt{2k}\). The value
\[P = 1 - \int_0^{T}f(x;k)\dd{x}\] states with which probability the
\(T\) value would be greater than the obtained one, where \(f\) is the
probability density of the \(\chi^2\) distribution. Thus
\(P\in [0,1]\) is a measure of confidence for the compatibility of the
histograms.
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "../document"

3
latex/tex/introduction.tex
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@ -52,6 +52,9 @@ Throughout natural units with
otherwise. The fine structure constant's value \(\alpha = 1/137.036\)
is configured in \sherpa\ and used in analytic calculations.
The compatibility of histograms is tested as discussed in
\cref{sec:comphist}.
\section{Source Code}%
\label{sec:source}

3
latex/tex/monte-carlo/sampling.tex
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@ -355,7 +355,8 @@ is a singularity at \(\pt = \ecm\), due to a term
\(1/\sqrt{1-(2\cdot \pt/\ecm)^2}\) stemming from the Jacobian
determinant. This singularity will vanish once considering a more
realistic process (see \cref{chap:pdf}). Furthermore the histograms
\cref{fig:histeta,fig:histpt} are consistent with their
\cref{fig:histeta,fig:histpt} have a \(P\)-value (see
\cref{sec:comphist}) tested for consistency with their
\rivet-generated counterparts and are therefore considered valid.
%%% Local Variables:

92
latex/tex/pdf/results.tex
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@ -92,51 +92,65 @@ being very steep.
%
To remedy that, one has to use a more efficient sampling algorithm
(\vegas) or impose very restrictive cuts. The self-coded
implementation used here can be found in \cref{sec:pycode} and employs
stratified sampling (as discussed in \cref{sec:stratsamp-real}) and
the hit-or-miss method. The matrix element (ME) and cuts are
implemented using \texttt{cython}~\cite{behnel2011:cy} to obtain
better performance as these are evaluated very often. The ME and the
cuts are then convolved with the PDF (as in \cref{eq:weighteddist})
and wrapped into a simple function with a generic interface and
plugged into the \vegas\ implementation which then computes the
integral, grid, individual contributions to the grid and rough
estimates of the maxima in each hypercube. In principle the code could
be generalized to other processes by simply redefining the matrix
elements, as no other part of the code is process specific. The cuts
work as simple \(\theta\)-functions, which has the advantage, that the
maximum for hit or miss can be chosen with respect to those cuts. On
the other hand, this method introduces discontinuity into the
integrand, which is problematic for numeric maximizers. The estimates
of the maxima, provided by the \vegas\ implementation used as the
starting point for a gradient ascend maximizer. In this way, the
discontinuities introduced by the cuts got circumvented. Because the
stratified sampling requires very accurate upper bounds, they have
been overestimated by \result{xs/python/pdf/overesimate}\!, which
lowers the efficiency slightly but reduces bias. The sampling
algorithm chooses hypercubes randomly in accordance to their
contribution to the integral by generating a uniformly distributed
random number \(r\in [0,1]\) and summing the weights of the hypercubes
until the sum exceeds this number. The last hypercube in this sum is
then chosen and one sample is obtained. Taking more than one sample
can improve performance, but introduces bias, as hypercubes with low
weight may be oversampled. At various points, the
\texttt{numba}~\cite{lam2015:po} package has been used to just-in-time
compile code to increase performance. The python
\texttt{multiprocessing} module is used to parallelize the sampling
and exploit all CPU cores. Although the \vegas\ step is very
(\emph{very}) time intensive, but the actual sampling performance is
in the same order of magnitude as \sherpa, but some parameters have to
be manually tuned.
implementation used here can be found as described in
\cref{sec:source} and employs stratified sampling (as discussed in
\cref{sec:stratsamp-real}) and the hit-or-miss method. The matrix
element (ME) and cuts are implemented using
\texttt{cython}~\cite{behnel2011:cy} to obtain better performance as
these are evaluated very often. The ME and the cuts are then convolved
with the PDF (as in \cref{eq:weighteddist}) and wrapped into a simple
function with a generic interface and plugged into the \vegas\
implementation which then computes the integral, grid, individual
contributions to the grid and rough estimates of the maxima in each
hypercube. In principle the code could be generalized to other
processes by simply redefining the matrix elements, as no other part
of the code is process specific. The cuts work as simple
\(\theta\)-functions, which has the advantage, that the maximum for
hit or miss can be chosen with respect to those cuts. On the other
hand, this method introduces discontinuity into the integrand, which
is problematic for numeric maximizers. The estimates of the maxima,
provided by the \vegas\ implementation used as the starting point for
a gradient ascend maximizer. In this way, the discontinuities
introduced by the cuts got circumvented. Because the stratified
sampling requires very accurate upper bounds, they have been
overestimated by \result{xs/python/pdf/overesimate}\!, which lowers
the efficiency slightly but reduces bias. The sampling algorithm
chooses hypercubes randomly in accordance to their contribution to the
integral by generating a uniformly distributed random number
\(r\in [0,1]\) and summing the weights of the hypercubes until the sum
exceeds this number. The last hypercube in this sum is then chosen and
one sample is obtained. Taking more than one sample can improve
performance, but introduces bias, as hypercubes with low weight may be
oversampled. At various points, the \texttt{numba}~\cite{lam2015:po}
package has been used to just-in-time compile code to increase
performance. The python \texttt{multiprocessing} module is used to
parallelize the sampling and exploit all CPU cores. Although the
\vegas\ step is very (\emph{very}) time intensive, but the actual
sampling performance is in the same order of magnitude as \sherpa, but
some parameters have to be manually tuned.
A sample of \result{xs/python/pdf/sample_size} events has been
generated both in \sherpa\ (with the same cuts) and through own
code. The resulting histograms of some observables are depicted in
\cref{fig:pdf-histos}. The sampling efficiency achieved was
\result{xs/python/pdf/samp_eff} using a total of
\result{xs/python/pdf/num_increments} hypercubes. The distributions
are compatible with each other. The sherpa runcard utilized here and
the analysis used to produce the histograms can be found in
\result{xs/python/pdf/num_increments} hypercubes.
The distributions are more or less compatible with each other
\footnote{See \cref{sec:comphist} for a description of the
compatibility test.}. In all cases the difference between
\(T\)-Value and the mean of the \(\chi^2\) distribution for that value
(\(=50\), the number of bins) is less then the standard deviation
(\(=10\)) of the same distribution and thus the histograms are
considered compatible. The very steep distributions for \(\pt\) and
\(m_{\gamma\gamma}\) are especially sensitive to fluctuations and the
systemic errors introduced of the weight of each hypercube. Therefore
their formal measure of compatibility, the \(P\)-Value, is rather
low. This shows that the error in the determination of the weights for
the hypercubes should be studied more carefully.
The \sherpa\ runcard utilized here and the analysis used to produce
the histograms can be found in
\cref{sec:ppruncard,sec:ppanalysis}. When comparing
\cref{fig:pdf-eta,fig:histeta} it becomes apparent, that the PDF has
substantial influence on the resulting distribution. Also the center

31
latex/tex/pheno.tex
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@ -3,26 +3,27 @@
\label{chap:pheno}
In real proton scattering the hard process discussed in
\cref{chap:pdf} is but only a part of the whole picture. Partons do in
\cref{chap:pdf} is only a part of the whole picture. Partons do in
general have some intrinsic transverse momentum. Scattered charges
radiate in both QCD and QED, the former radiation giving rise to
parton-showers and additional transverse momentum of the partons. The
remnants of the proton can radiate showers themselves, scatter in more
or less hard processes (Multiple Interactions, MI) and affect the hard
process through color correlation. All of the processes not directly
connected to the hard process are called the underlying event and have
to be taken into account to generate events that can be compared with
experimental data. Finally the partons from the showers recombine into
hadrons (Hadronization) due to confinement. This last effect doesn't
radiate in both QCD and QED, both giving rise to shower-like cascades
and both can lead to additional transverse momentum of the initial
state partons. The remnants of the proton can radiate showers
themselves, scatter in more or less hard processes (Multiple
Interactions, MI) and affect the hard process through color
correlation. All of the processes not directly connected to the hard
process are called the underlying event and have to be taken into
account to generate events that can be compared with experimental
data. Finally the partons from the showers recombine into hadrons
(hadronization) due to QCD confinement. This last effect doesn't
produce diphoton-relevant background directly, but affects photon
isolation.~\cite[11]{buckley:2011ge} % TODO: describe isolation
isolation.~\cite[11]{buckley:2011ge}
These effects can be calculated or modeled on an per-event base by
modern monte-carlo event generators like \sherpa\footnote{But these
calculations and models are always approximations.}. This is done
modern Monte Carlo event generators like \sherpa. But these
calculations and models are approximations in most cases. This is done
for the diphoton process in a gradual way described in
\cref{sec:setupan}. Histograms of observables are generated and are
being discussed in \cref{sec:disco}.
\cref{sec:setupan}. Histograms of observables are generated and
discussed in \cref{sec:disco}.
%%% Local Variables:
%%% mode: latex

132
latex/tex/pheno/discussion.tex
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@ -6,6 +6,19 @@
\rivethist{pheno/xs}
\caption{\label{fig:disc-xs}}
\end{subfigure}
\begin{subfigure}[t]{.49\textwidth}
\rivethist{pheno/isolation_discard}
\caption{\label{fig:disc-iso-disc}}
\end{subfigure}
\begin{subfigure}[t]{.49\textwidth}
\rivethist{pheno/cut_discard}
\caption{\label{fig:disc-cut-disc}}
\end{subfigure}
\caption{Cross section and event discard statistics plots.}
\end{figure}
\begin{figure}[ht]
\centering
\begin{subfigure}[t]{.49\textwidth}
\rivethist{pheno/cos_theta}
\caption{\label{fig:disc-cos_theta}}
@ -26,7 +39,6 @@
\rivethist{pheno/o_angle}
\caption{\label{fig:disc-o_angle}}
\end{subfigure}
\caption{Continued on next page.}
\end{figure}
%
\begin{figure}[t]
@ -52,50 +64,65 @@
by simulations with increasingly more effects turned on.}
\end{figure}
%
The results of the \sherpa\ runs for each stage with \(10^6\) events
The results of the \sherpa\ runs for each stage with \(10^7\) events
each are depicted in the histograms in \cref{fig:holhistos} and shall
now be discussed in detail.
%TODO: high prec not possible
Because of the analysis cuts, the total number of accepted events is
smaller than the number of events generated by \sherpa, but sufficient
as can be seen. The fiducial cross sections of the different stages,
which are compared in \cref{fig:disc-xs}, differ as a result of
that. All other histograms are normalized to their respective cross
sections.
for proper statistics for most observables. The fiducial cross
sections of the different stages, which are compared in
\cref{fig:disc-xs}, differ as a result.
% TODO: not as result
All other histograms
are normalized to their respective cross sections.
Effects that give the photon system additional $\pt$ decrease the
cross section. This can be understood as follows. When there is no
additional \(\pt\), then the photon momenta are back to back in the
plane perpendicular to the beam axis. If the system now gets a kick
then this usually subtracts \(\pt\) from one of the photons unless
that kick is perpendicular to the photons. Because the \(\pt\)
distribution (\cref{fig:disc-pT,fig:disc-pT_subl}) is very steep, a
lot of events produce photons with low \(\pt\) and so this effect is
substantial. The isolation cuts do affect the cross section as well
The \stfour\ cross section is a bit higher than the \stthree\ one,
because the harmonization favors isolation of photons by reducing the
number of particles in the final state. The opposite effect can be
seen with MI, where the number of final state particles is increased.
plane perpendicular to the beam axis (transverse plane). If the system
now gets a kick then this usually subtracts \(\pt\) from one of the
photons unless that kick is near perpendicular to the photons. Because
the \(\pt\) distribution (\cref{fig:disc-pT,fig:disc-pT_subl}) is very
steep, a lot of events produce photons with low \(\pt\) and so this
effect is substantial. The fraction of events that have been discarded
by the \(\eta\) and \(\pt\) cuts are plotted in
\cref{fig:disc-cut-disc}, which shows an increase for all stages after
\stone, leading (principally) to the drop in cross section for the
\sttwo\ and \stthree.
The isolation cuts do affect the cross section as well, as is
demonstrated in \cref{fig:disc-iso-disc} which shows the fraction of
events discarded due to the isolation cuts. The \stfour\ cross section
is a bit higher than the \stthree\ one, because the hardonization
favors isolation of photons by reducing the number of particles in the
final state and clustering them closer together. The opposite effect
can be seen with MI, where the number of final state particles is
increased and this effect leads to another substantial drop in the
cross section.
% TODO: analysis plot of rejected events?
% TODO: link to CS frame
% TODO: iso cuts
% TODO: hadr isolation? why
% TODO: teilchen aufgefaechert, weniger in cone, teilchen ohne calo
The transverse momentum of the photon system (see
\cref{fig:disc-total_pT}) now becomes non trivial, as both the \sttwo\
and \stthree stage affect this observable directly. Initial state
and \stthree\ stage affect this observable directly. Initial state
radiation generated by the parton showering algorithm kicks the quarks
involved in the hard process and thus generates transverse
momentum. In regions of high \(\pt\) all but the \stone\ stage are
largely compatible, falling off steeply at
involved in the hard process and thus generates transverse momentum
and primordial \(\pt\) is simulated by the \stthree\ stage. In regions
of high \(\pt\) all but the \stone\ stage are largely compatible,
falling off steeply at
\(\mathcal{O}(\SI{10}{\giga\electronvolt})\). In the region of
\SI{1}{\giga\electronvolt} and below, the effects primordial \(\pt\)
show as an enhancement in cross section. This is consistent with the
mean of the primordial \(\pt\) distribution which was off the order of
\gev{1}. The distribution for MI is enhanced at very low \(\pt\) which
could be an isolation effect or stem from the fact, that other partons
can be showers as well decreasing the showering probability for the
partons involved in the hard scattering.
can emit QCD bremsstrahlung and showers as well, decreasing the
showering probability for the partons involved in the hard scattering.
% TODO: clarify, Frank
The fact that the distribution has a maximum and falls off towards
lower \(\pt\) relates to the fact, that parton shower algorithms
@ -108,7 +135,7 @@ back to back photons are favored by all distributions and most events
feature an azimuthal separation of less than \(\pi/2\), the
enhancement of the low \(\pt\) regions in the \stthree\ stage also
leads to an enhancement in the back-to-back region for this stage over
the \stone\ stage.
the \sttwo\ stage.
In the \(\pt\) distribution of the leading photon (see
\cref{fig:disc-pT}) the boost of the leading photon towards higher
@ -118,20 +145,23 @@ compatible beyond \gev{1}. Again, the effect of primordial \(\pt\)
becomes visible transverse momenta smaller than \gev{1}.
% TODO: mention steepness again
The \(\pt\) distribution for the subleading photon shows remarkable
The \(\pt\) distribution for the sub-leading photon shows remarkable
resemblance to the \stone\ distribution for all other stages, although
there is a very minute bias to lower \(\pt\). This is consistent with
the mechanism described above so that events that subtract \(\pt\)
from the subleasing second photon are favored. Interestingly, the
effects of primordial \(\pt\) not very visible at all.
the mechanism described above so that events that subtract (very small
amounts of) \(\pt\) from the sub-leading second photon are more
common. Interestingly, the effects of primordial \(\pt\) not very
visible.
The distribution for the invariant mass (see \cref{fig:disc-inv_m})
shows that events with lower c.m.\ energies than in the \stone\ can
pass the cuts by being \(\pt\) boosted although. The decline of the
cross section towards lower energies is much steeper than the decline
towards higher energies, which originates from the PDFs. The tendency
for higher \(\pt\) boosts of the photon system shows in a slight
enhancement of the \sttwo cross section in the \sttwo\ cross section.
shows that events with lower c.m.\ energies than the \stone\ threshold
can pass the cuts by being \(\pt\) boosted. The decline of the cross
section towards lower energies is much steeper than the PDF-induced
decline towards higher energies. High \(\pt\) boost to \emph{both}
photons are very rare, which supports the reasoning about the drop in
total cross section. The tendency for higher \(\pt\) boosts of the
photon system in the \sttwo\ stage shows in a slight enhancement of
the \sttwo\ cross section at low \(\pt > \gev{2}\).
The angular distributions of the leading photon in
\cref{fig:disc-cos_theta,fig:disc-eta} are most affected by the
@ -139,26 +169,26 @@ differences in total cross section and slightly shifted towards more
central (transverse) regions for all stages from \sttwo\ on due to the
\(\pt\) kicks to the photon system.
Because the diphoton process itself is only affected by kinematic
changes to the initial change quarks, the scattering angle cross
sections in \cref{fig:disc-o_angle,fig:disc-o_angle_cs} show a similar
shape in all stages. Towards small scattering angles, the differences
in shape grow larger, as this is the region where the cuts have the
largest effect. In the CS frame, the cross section does not converge
to zero for \sttwo\ and subsequent stages. With non-zero \(\pt\) of
the photon system, the z-axis of the CS frame rotates out of the
region that is affected by cuts. The ration plot also shows, that the
region where cross section distributions are similar in shape extends
further. In the CS frame effects of the non-zero \(\pt\) of the photon
system are (somewhat weakly) suppressed.
Because the diphoton system itself is only affected by recoil to the
initial state quarks, the scattering angle cross sections in
\cref{fig:disc-o_angle,fig:disc-o_angle_cs} show a similar shape in
all stages. Towards small scattering angles, the differences in shape
grow larger, as this is the region where the cuts have the largest
effect. In the CS frame, the cross section does not converge to zero
for \sttwo\ and subsequent stages. With non-zero \(\pt\) of the photon
system, the z-axis of the CS frame rotates out of the region that is
affected by cuts. The ration plot also shows, that the region where
cross section distributions are similar in shape extends further. In
the CS frame effects of the non-zero \(\pt\) of the photon system are
(somewhat weakly) suppressed.
It becomes clear, that the \sttwo\ and \stthree\ have the biggest
effect on the shape of observables, as they affect kinematics
directly. Isolation effects show most with the \stfour\ and \stfive\
stages. In angular observables hard process alone seems to be a
reasonable good approximation, but in most other observables non-LO
effects introduce considerable deviations and have to be taken into
account.
directly. Isolation effects show most with the \stfour\ and especially
the \stfive\ stages. In angular observables hard process alone gives a
reasonably good qualitative picture, but in most other observables
non-LO effects introduce considerable deviations and have to be taken
into account.
%%% LOCAL Variables:
%%% mode: latex

51
latex/tex/pheno/setup.tex
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@ -4,22 +4,30 @@
To observe the impact on the individual aspect of the proton
scattering the following run configurations have been defined. They
are incremental in the sense that each subsequent configuration
extents the previous one and thus called stages.
extents the previous one and thus called stages from now on and are
listed below.
%
\begin{description}
\item[LO] The hard process on parton level as used in \cref{sec:pdf_results}.
\item[LO+PS] The shower generator of \sherpa, \emph{CSS} (dipole-shower),
is activated and simulates initial state radiation, as there are no
partons in the final state yet.
\item[LO+PS+pT] The beam remnants are simulated, giving rise to final state radiation.
Also the partons are being assigned primordial \(\pt\), distributed
like a Gaussian with a mean value of \SI{.8}{\giga\electronvolt} and
a standard deviation of \SI{.8}{\giga\electronvolt}\footnote{Those
values are \sherpa 's defaults.}.
\item[LO+PS] The shower generator of \sherpa, \emph{CSS}
(dipole-shower), is activated and simulates initial state
radiation. The recoil scheme proposed in~\cite{hoeche2009:ha}, which
has been proven more accurate for diphoton production at leading
order, has been enabled.
\item[LO+PS+pT] The beam remnants are simulated, giving rise to
aditional radiation and parton showers. Also the partons are being
assigned primordial \(\pt\), distributed like a Gaussian with a mean
value of \SI{.8}{\giga\electronvolt} and a standard deviation of
\SI{.8}{\giga\electronvolt}\footnote{Those values are \sherpa 's
defaults.}.
\item[LO+PS+pT+Hadronization] A cluster hadronization model
implemented in \emph{Ahadic} is activated.
\item[LO+PS+pT+Hadronization+MI] Multiple interactions based on the
Sj\"ostrand-van-Zijl Model are simulated.
implemented in \emph{Ahadic} is activated. The shower particles are
being hadronized and the decay of the resulting hadrons simulated if
they are unstable.
\item[LO+PS+pT+Hadronization+MI] Multiple Interactions (MI) based on
the Sj\"ostrand-van-Zijl Model are simulated. The MI are parton
shower corrected, so that there are generally more particles in the
final state.
\end{description}
%
A detailed description of the implementation of those models can be
@ -41,23 +49,28 @@ non-zero transverse momentum of the photon pair:
%
\begin{itemize}
\item total transverse momentum of the photon pair
\item azimuth angle between the two photons
\item azimuthal angle between the two photons
\item transverse momentum of the sub-leading photon (see below)
\end{itemize}
%
Because the final state now potentially contains additional photons
from hadron decays, the analysis only selects prompt photons with the
highest \(\pt\) (leading photons). Furthermore a cone of
\(R = \sqrt{\qty(\Delta\varphi)^2 + \qty(\Delta\eta)^2} = 0.4\) around
each photon must not contain more than \SI{4.5}{\percent} of the
photon transverse momentum (\(+ \SI{6}{\giga\electronvolt}\)),
\[R = \sqrt{\qty(\Delta\varphi)^2 + \qty(\Delta\eta)^2} \leq 0.4\]
around each photon must not contain more than \SI{4.5}{\percent} of
the photon transverse momentum (\(+ \SI{6}{\giga\electronvolt}\)),
attempting to exclude photons stemming from hadron decay are filtered
out. The leading photons are required to have \(\Delta R > 0.45\), to
filter out colinear photons, as they likely stem from hadron
decays. In truth, the analysis already excludes such photons, but to
decays.
% TODO: only for experiments, do not overlap photon iso cones, einfach
% weglassen
In truth, the analysis already excludes such photons, but to
be compatible with experimental data, which must rely on such
criteria, they have been included. The code of the analysis is listed
in \cref{sec:ppanalysisfull}.
criteria, they have been included. These cuts are called
\emph{isolation cuts}. The code of the analysis is listed in
\cref{sec:ppanalysisfull}.
The production of photons in showers has not been considered.